Problem: Let $F$ be a vector field. Is the expression $\nabla \cdot (\nabla \cdot F)$ a scalar field, a vector field, or undefined? Choose 1 answer: Choose 1 answer: (Choice A) A Scalar field (Choice B) B Vector field (Choice C) C Undefined
Solution: The divergence, which takes a vector field and gives a scalar field, can be written in two ways: $\text{div}(F) = \nabla \cdot F$ Therefore, $\nabla \cdot (\nabla \cdot F)$ is the divergence of the divergence of a vector field. The divergence of a vector field is a scalar field. Because the divergence only takes vector fields, the divergence of that scalar field is undefined. The expression $\nabla \cdot (\nabla \cdot F)$ is undefined.